Domain and range of \arccos x?

Domain -1\le x\le1; range 0\le y\le\pi.

Domain and range of \arctan x?

Domain all real x; range -\frac{\pi}{2}<y<\frac{\pi}{2}.

How is the graph of an inverse function related to the original?

It is the reflection of the restricted original in the line y=x.

Why do sin, cos and tan need restricted domains before they have inverses?

They are periodic, so not one-to-one; restricting makes each output come from one input.

What is \arcsin(\sin150^{\circ})?

30^{\circ}, because \arcsin returns an angle in [-90^{\circ},90^{\circ}].

What are the horizontal asymptotes of y=\arctan x?

y=\frac{\pi}{2} and y=-\frac{\pi}{2}.

No, because 2 is outside the domain [-1,1].

Is \sin^{-1}x the same as \frac{1}{\sin x}?

No. \sin^{-1}x=\arcsin x is the inverse function; \frac{1}{\sin x}=\operatorname{cosec}x.

Inverse trigonometric functionsEdexcel International A Level Maths: Flashcards

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Domain and range of $\arcsin x$?

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Domain and range of $\arcsin x$ ?: Domain $-1\le x\le1$ ; range $-\frac{\pi}{2}\le y\le\frac{\pi}{2}$ .
Domain and range of $\arccos x$ ?: Domain $-1\le x\le1$ ; range $0\le y\le\pi$ .
Domain and range of $\arctan x$ ?: Domain all real $x$ ; range $-\frac{\pi}{2}<y<\frac{\pi}{2}$ .
How is the graph of an inverse function related to the original?: It is the reflection of the restricted original in the line $y=x$ .
Why do sin, cos and tan need restricted domains before they have inverses?: They are periodic, so not one-to-one; restricting makes each output come from one input.
Exact value of $\arcsin\frac12$ ?: $\frac{\pi}{6}$ ( $30^{\circ}$ ).
Exact value of $\arccos\left(-\frac12\right)$ ?: $\frac{2\pi}{3}$ ( $120^{\circ}$ ).
Exact value of $\arctan\sqrt3$ ?: $\frac{\pi}{3}$ ( $60^{\circ}$ ).
What is $\arcsin(\sin150^{\circ})$ ?: $30^{\circ}$ , because $\arcsin$ returns an angle in $[-90^{\circ},90^{\circ}]$ .
When is $\arccos(\cos x)=x$ ?: Only when $0\le x\le\pi$ .
What are the horizontal asymptotes of $y=\arctan x$ ?: $y=\frac{\pi}{2}$ and $y=-\frac{\pi}{2}$ .
Does $\arcsin2$ exist?: No, because $2$ is outside the domain $[-1,1]$ .
Is $\sin^{-1}x$ the same as $\frac{1}{\sin x}$ ?: No. $\sin^{-1}x=\arcsin x$ is the inverse function; $\frac{1}{\sin x}=\operatorname{cosec}x$ .

Exam questions on Inverse trigonometric functions

The inverse trigonometric functions $\arcsin$ , $\arccos$ and $\arctan$ are defined by their principal values, with angles in radians.
Find the exact value of $\arctan\left(-\sqrt3\right)+\arctan(1)$ .2 marks
Let $\alpha=\arcsin\left(\frac{4}{5}\right)$ , where $\alpha$ is measured in degrees.
Find $\arcsin(\sin150^{\circ})$ and explain why it is not $150^{\circ}$ .2 marks
Angles are measured in radians and a calculator may be used.
Solve $\tan x=-3$ for $-\pi<x<\pi$ , giving your answers to 3 significant figures.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).